Add one meter to a rope that fits snugly around the Earth’s equator, lift it evenly all the way round, and the gap underneath is 1/(2π) meters: about 15.9 centimeters, enough for a mouse to walk under. The size of the Earth does not matter. Do the same with a rope around a basketball and the gap is the same 15.9 centimeters, because the radius cancels out of the sums.
Almost everyone’s first guess is “far too small to notice”. The equator is about 40,000 kilometers long, and one meter is one part in forty million of that. Spread over the whole planet, the extra meter should vanish. It doesn’t.
The setup
Treat the Earth as a perfect sphere and the rope as a circle around its equator. Call the Earth’s radius r, and the gap between the ground and the lengthened rope h, measured in meters. The supports in the video lift the longer rope by the same height everywhere, so the new circle is the old one with its radius grown by h.
Why the radius cancels
The circumference of a circle is 2π times its radius.
- The original rope: C = 2πr.
- The rope with one more meter: C + 1 = 2π(r + h) = 2πr + 2πh.
- Subtract the first line from the second. The C on the left and the 2πr on the right both disappear, leaving 1 = 2πh.
- So h = 1/(2π) meters.
Notice what is missing from the last line: r. The answer depends only on the extra length, never on the size of the circle. A planet, a basketball and a coin all give the same 15.9 centimeters. That is why the video ends on a basketball.
Why it feels wrong
Circumference and radius are tied by a fixed factor, 2π ≈ 6.28. Every meter of extra radius costs about 6.28 meters of extra rope, wherever the circle is and however big it is. Run that backwards: one extra meter of rope buys 1/6.28 of a meter of height, about 16 centimeters. Our intuition compares the extra meter with the whole equator, which makes it look tiny. The geometry compares it with 2π, which is not tiny at all.
Turn the question around and it is just as strange. To lift a rope one full meter off the ground all the way round the Earth, you need only 2π, about 6.3, extra meters of rope, not thousands of kilometers.
An old puzzle
The puzzle is more than three centuries old. A version of it, “or one very like it”, in the careful words of David Darling’s encyclopedia, appeared in a students’ book on Euclid by William Whiston, the English clergyman and mathematician who succeeded Isaac Newton as Lucasian professor at Cambridge in May 1702. The book was Whiston’s edition of Andrew Tacquet’s Euclid, written for young students at the university; Darling dates it to 1702, MacTutor to the following year. The puzzle has been catching people out ever since.
Long problems often hide a short answer, and this one hides it in a subtraction. For another puzzle where a simple count beats intuition, see why 31 dominoes can’t cover a cut chessboard, and for more on why 2π is the natural measure of a turn, π as the unit of turning.
One extra meter, any circle: about 16 centimeters.
